Much less has been done for the case of non-zero force field. For instance, all works we metioned above, except [1]}, deal with the diffusion without a force field, that is \(F=0\) . We could find two works concerning this case: [2]} and [1]}. Both these works consider \(M\)  being a domain in \(\mathbb {R}^2\)  or \(\mathbb {R}^3\) . In [2]}, the authors generalize the method of [5]}, [6]}, [7]} to obtain the leading-order term of the average of the expected first arrival time for two and three dimensional cases. For planar domains, the authors in [1]}, by using layer potential techniques, derive assymptotic expansion up to \(O(\varepsilon )\)  term.
